The proposed research aims (1) to develop new methods of mathematical description of growth, (2) to test their utility in a number of specific biological cases, including human and rodent craniofacial growth and molluscan shell growth, (3) to develop and integrate descriptions of the underlying morphodynamic processes with the kinematic modeling, and (4) to explore clinical and phylogenetic applications. The mathematical modeling is based on and extends concepts developed in the mechanics of continua. Growth is described by mass sources distributed volumetrically and over certain growth surfaces. The results may be summarized by allometric relations between certain finite dimensions. An allometric network model is postulated for various parts of the skull. Statistical testing of hypotheses against biologic data and exhibition of the results by computer graphics is a major emphasis of the research. The biological cases in which the modeling will be tested include pre- and postnatal human cephalic skeletal and visceral growth. Additionally, the rat provides a unique normal and abnormal cephalogenesis. Normally, there is a marked relative motion of the facial and neural skulls; and experimental hydro (macro) cephaly and cerebral and cerebellar microcephaly can be dependably produced. These rat models permit integration of underlying morphodynamic processes to the kinematic modeling, another major emphasis of the research. The mathematical and computational methods proposed will have wide applicability to various growth situations: to the clinical analysis of both normal and abnormal cephalic growth patterns and in othodontic prediction and interpretation of growth patterns generally. The research will provide also a kinematic basis for phylogenetic size and shape transformations.